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arXiv · 1709.07939

Ideal membership in $H^\infty$: Toeplitz corona approach

Michael Hartz·Brett D. Wick
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Abstract

We study the ideal membership problem in $H^\infty$ on the unit disc. Thus, given functions $f,f_1,\ldots,f_n$ in $H^\infty$, we seek sufficient conditions on the size of $f$ in order for $f$ to belong to the ideal of $H^\infty$ generated by $f_1,\ldots,f_n$. We provide a different proof of a theorem of Treil, which gives the sharpest known sufficient condition. To this end, we solve a closely related problem in the Hilbert space $H^2$, which is equivalent to the ideal membership problem by the Nevanlinna-Pick property of $H^2$.

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Michael Hartz, Brett D. Wick. 2017-09-22. Ideal membership in $H^\infty$: Toeplitz corona approach. https://doi.org/10.1007/s00020-018-2488-9

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